Identification of Quantum Injective Dual Frames on \(\mathbb {R}^n\)

The quantum injectivity problem classifies frames which are injective with respect to self-adjoint Hilbert-Schmidt operators. In this paper, we aim to analyze quantum injective frames in terms of the excess of frame elements in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ . Especially, we investigate the connection between the injectivity, full spark and phase retrieval frames. In addition, we detect injective (alternate) dual frames and show that the family of quantum injective dual frames is open and dense in the set of all dual frames. Finally, the stability of quantum injective frames will be addressed.